About tangent estimators Better ideas ? Everything is cool, but… Can we do better than using digital straight segments ? Not only for tangent estimation, but also for conversion from raster to vector graphics. Use digital primitives of higher degree . 44
About tangent estimators Better ideas ? Curvature is defined with An analytical function is approximated by osculating circles its Taylor Polynomial of degree n. Use digital circles Use a more generic approach Use digital primitives of higher degree . 45
About tangent estimators Better ideas ? Use a more generic approach 46
Plan Introduction About tangent estimators Digital Level Layers DLL decomposition Algorithm 47
Plan Digital Level Layers 48
Digital Level Layers Warning Usual geometry is based on real numbers, which by paradox are ’’ unreal ’’. World of Reals. limit Numbers with a finite description There is not only one way to and a finite time… discretize a real concept… Different discrete objects or concepts have the same limit …
Digital Level Layers Approaches Continuous figures. Three approaches can be used to define digital primitives: - topological - morphological - analytical Digital figures.
Digital Level Layers Illustration on an ellipse Task: define a digital primitive for S.
Digital Level Layers Topological ellipse Task: define a digital primitive for S.
Minkowski’s sum Digital Level Layers The Minkowski’s sum S+B is the set of points covered by the structuring elements as it moves all along the shape. A shape S A structuring element B 53
Minkowski’s sum Digital Level Layers The Minkowski’s sum S+B is the set of points covered by the structuring elements as it moves all along the shape. A shape S A structuring element B The dilation of S by B 54
Digital Level Layers Morphological ellipse Structuring element
Digital Level Layers Analytical ellipse We relax the equality f(x)=h in a double inequality h- Δ/2 ≤ f(x)<h + Δ/2.
Digital Level Layers 3 digital ellipses Topological approach. Morphological approach. Analytical approach. The 3 definitions collapse for lines in Z² , planes in Z 3 … hyperplanes in Z d (affine sub-spaces of codimension 1 ) Each approach has its own parameters but there is a correspondance. Topology Morphology Algebra value Δ Neighborhood Structuring element Ball N ∞ Ball N 1 Δ =N ∞ (a) Ball N 1 Δ =N 1 (a) Ball N ∞
Digital Level Layers Naïve objects Topological approach. Morphological approach. Analytical approach. The 3 definitions collapse for lines in Z² , planes in Z 3 … hyperplanes in Z d (affine sub-spaces of codimension 1 ) Each approach has its own parameters but there is a correspondance. Topology Morphology Algebra Naïve class. value Δ Neighborhood Structuring element Ball N ∞ Ball N 1 Δ =N ∞ (a) Ball N 1 Δ =N 1 (a) Ball N ∞
Digital Level Layers Standard objects Topological approach. Morphological approach. Analytical approach. The 3 definitions collapse for lines in Z² , planes in Z 3 … hyperplanes in Z d (affine sub-spaces of codimension 1 ) Each approach has its own parameters but there is a correspondance. Topology Morphology Algebra value Δ Neighborhood Structuring element Standard class. Ball N ∞ Ball N 1 Δ =N ∞ (a) Ball N 1 Δ =N 1 (a) Ball N ∞
Digital Level Layers Bad point The 3 definitions collapse for lines in Z² , planes in Z 3 … hyperplanes in Z d (affine sub-spaces of codimension 1 ) They don’t collapse for arbitrary shapes.
Digital Level Layers Good and bad points Topology Morphology Analysis Properties Topology Morphology Algebraic characterization Recognition algorithm
Digital Level Layers Good and bad points Topology Morphology Analysis Properties Topology Morphology Algebraic characterization Recognition algorithm SVM
Digital Level Layers Aïe Analysis Topology Morphology
Digital Level Layers Definition Analysis Topology Morphology Digital Level Layer definition: A Digital Level Layer (name coming from Level sets ) is a subset of Z d characterized by a double inequality: h ≤ f(x) < h’
Digital Level Layers Sphere Digital Level Layer (DLL for short) Digital Level Layer definition: A Digital Level Layer (name coming from Level sets ) is a subset of Z d characterized by a double inequality: h ≤ f(x) < h’
Digital Level Layers Hyperboloïd Digital Level Layer (DLL for short) Digital Level Layer definition: A Digital Level Layer (name coming from Level sets ) is a subset of Z d characterized by a double inequality: h ≤ f(x) < h’
Digital Level Layers From raster to vector graphics The advantage of DLL is that they are described by double-inequalities: They can be used in Vector Graphics (for zooming or any transformation). Digital Level Layer (DLL for short) Digital Level Layer definition: A Digital Level Layer (name coming from Level sets ) is a subset of Z d characterized by a double inequality: h ≤ f(x) < h’
Digital Level Layers Theoretical results ? Digital Level Layers generalize Digital Straight Lines. What about Tangent estimations and multigrid convergence? Digital Level Layer definition: A Digital Level Layer (name coming from Level sets ) is a subset of Z d characterized by a double inequality: h ≤ f(x) < h’
Digital Level Layers Derivatives estimators Review of multigrid convergent estimators developped in Digital Geometry. Assumption Order Worst case Method on the continuous Authors of Error bound curve derivative A. Vialard, Maximal Locally convex, k=1 O(h 1/3 ) J-O Lachaud, DSS with C 3 thickness=1 F De Vieilleville S. Fourey, F. Brunet, k O(h (2/3) ) A. Esbelin, C 3 or C 2 Any k Convolutions B. R. Malgouyres Maximal L. Provot, O(h (1/(k+1)) ) DLL with C k+1 Any k Y. Gerard thickness>1
Digital Level Layers Derivatives estimators Review of multigrid convergent estimators developped in Digital Geometry. Assumption Order Worst case Method on the continuous Authors of Error bound curve derivative A. Vialard, Maximal Locally convex, k=1 O(h 1/3 ) J-O Lachaud, DSS with C 3 thickness=1 F De Vieilleville S. Fourey, F. Brunet, k A. Esbelin, O(h (2/3) ) C 3 or C 2 Any k Convolutions B. R. Malgouyres Parameter free because the parameter Maximal i.e the class of digital L. Provot, DLL with O(h (1/(k+1)) ) C k+1 Any k straight lines has been Y. Gerard thickness>1 fixed …
Digital Level Layers Derivatives estimators Review of multigrid convergent estimators developped in Digital Geometry. All approaches are able to deal with noisy shapes Assumption Order Worst case (using their parameters). Method on the continuous Authors of Error bound curve derivative A. Vialard, Maximal Locally convex, k=1 O(h 1/3 ) J-O Lachaud, DSS with C 3 thickness=1 F De Vieilleville S. Fourey, F. Brunet, k O(h (2/3) ) A. Esbelin, C 3 or C 2 Any k Convolutions B. R. Malgouyres Maximal L. Provot, O(h (1/(k+1)) ) DLL with C k+1 Any k Y. Gerard thickness>1
Digital Level Layers Difference Review of multigrid convergent estimators developped in Digital Geometry. Can be applied on contours of a shape, not only the graph of a function … A. Vialard, Maximal Locally convex, k=1 O(h 1/3 ) J-O Lachaud, DSS with C 3 thickness=1 F De Vieilleville S. Fourey, F. Brunet, A. Esbelin, B. R. Malgouyres Maximal L. Provot, O(h (1/(k+1)) ) DLL with C k+1 Any k Y. Gerard thickness>1 Applied on a digital function f:Z → Z
Digital Level Layers Relations Review of multigrid convergent estimators developped in Digital Geometry. A. Vialard, Maximal Locally convex, k=1 O(h 1/3 ) J-O Lachaud, DSS with C 3 thickness=1 F De Vieilleville Better worst S. Fourey, F. Brunet, Increase weaker case More k A. Esbelin, O(h (2/3) ) C 3 or C 2 Any k Convolutions the assumption convergence general B. R. Malgouyres thickness rate. Maximal L. Provot, O(h (1/(k+1)) ) DLL with C k+1 Any k Y. Gerard thickness>1 Maximal There is the convergence result for k=1 . DLL with There should exist extensions for k>1 under some conditions… thickness=1
Digital Level Layers Relations Review of multigrid convergent estimators developped in Digital Geometry. Assumption Iterative version with deleting and points insertion for computing the derivative along a curve. Order Method on the continuous Authors It remains linear. of curve derivative Computation in worst case linear time for a single DLL. A. Vialard, Maximal Locally convex, k=1 O(h 1/3 ) J-O Lachaud, DSS with C 3 thickness=1 F De Vieilleville S. Fourey, F. Brunet, k A. Esbelin, O(h (2/3) ) C 3 or C 2 Any k Convolutions B. R. Malgouyres Maximal L. Provot, O(h (1/(k+1)) ) DLL with C k+1 Any k Y. Gerard thickness>1 Computation in O(n 2(k+1) ) in theory but close to linear time in practice for a single DLL. No iterative version with deleting and points insertion for computing the derivative along a curve. It becomes quadratic.
Digital Level Layers Relations Review of multigrid convergent estimators developped in Digital Geometry. More restrictive and less accurate but faster … A. Vialard, Maximal Locally convex, k=1 O(h 1/3 ) J-O Lachaud, DSS with C 3 thickness=1 F De Vieilleville S. Fourey, F. Brunet, k A. Esbelin, O(h (2/3) ) C 3 or C 2 Any k Convolutions B. R. Malgouyres Maximal L. Provot, O(h (1/(k+1)) ) DLL with C k+1 Any k Y. Gerard thickness>1
Digital Level Layers Relations Review of multigrid convergent estimators developped in Digital Geometry. Order Worst case Method Authors of Error bound derivative Maximal L. Provot, O(h (1/(k+1)) ) DLL with C k+1 Any k Y. Gerard thickness>1 How does it work ?
Digital Level Layers DLL for derivatives estimators We use DLL with double inequality: P(x) ≤ y < P(x)+ Δ with a fixed Δ >1 and a chosen maximal degree k for P(x) . If we choose a high Δ , it allows more noise, but becomes less precise. Δ y=P(x)+ Δ y=P(x) How does it work ?
Digital Level Layers DLL for derivatives estimators We use DLL with double inequality: P(x) ≤ y < P(x)+ Δ with a fixed Δ >1 and a chosen maximal degree k for P(x) . If we choose a high Δ , it allows more noise, but becomes less precise. Δ y=P(x)+ Δ y=P(x) P(x) provides directly the derivative of order k .
Second derivative
Digital Level Layers Multigrid convergence Second derivative
Plan Introduction About tangent estimators Digital Level Layers DLL decomposition Algorithm 84
Plan DLL decomposition 85
DLL Decomposition Undesired neighbors Input : A digital curve Segmentation in pieces of digital straight lines (72 pieces) Output : Its decomposition in Digital Straigh Segments
DLL Decomposition Undesired neighbors Principle : Digitization Undesired neighbors Recognition DLL containing S Input: Lattice set S
DLL Decomposition Inliers and outliers Principle : Digitization Undesired neighbors Recognition DLL containing S Input: Lattice set S + Recognition DLL between the inliers and outliers Forbidden neighbors
DLL Decomposition Examples Segmentation in pieces of Segmentation in pieces of Segmentation in pieces of digital straight lines digital circles (DLL) digital conics (DLL) (72 pieces) (24 pieces) (18 pieces) We decompose the digital curve in Digital Level Layers (DLL)
DLL Decomposition Examples Segmentation in pieces of Segmentation in pieces of Segmentation in pieces of digital straight lines digital circles (DLL) digital conics (DLL) (116 pieces) (50 pieces) (42 pieces) We decompose the digital curve in Digital Level Layers (DLL)
DLL Decomposition Examples It provides a vector description of a digital curve wich is smoother than DSS. All cases computed with a UNIQUE algorithm (with, as parameter, a chosen basis of polynomials like in SVM) Segmentation in pieces of Segmentation in pieces of Segmentation in pieces of digital straight lines digital circles (DLL) digital conics (DLL) (116 pieces) (50 pieces) (42 pieces)
DLL Decomposition IPOL Paper, Demo and code are available on IPOL (thanks to Bertrand Kerautret)
Plan Introduction About tangent estimators Digital Level Layers DLL decomposition Algorithm 93
Plan Algorithm 94
Algorithm Reduction to Linear separability Problem of separation Problem of linear separability by a level set f(x)=0 in a descriptive space of higher dimension with f in a given linear space Kernel trick (Aïzerman et al. 1964) is the principle of Support Vector Machines .
Algorithm GJK Problem of separation Problem of linear separability by a level set f(x)=0 in a descriptive space of higher dimension with f in a given linear space GJK ( Gilbert Johnson Keerthi , 1988) computes the closest pair of points from the two convex hulls. It’s widely used for collision detection.
Algorithm Variant with three sets of points Problem of separation Problem of linear separability by two level sets f(x)=h and f(x)=h’ by two parallel hyperplanes with f in a given linear space We introduce a variant of GJK in nD GJK ( Gilbert Johnson Keerthi , 1988) computes the closest pair of points from the two convex hulls. It’s widely used for collision detection.
Algorithm GJK Input : two polytopes A ⊂R d and B ⊂ R d given by their vertices. Question : do they intersect ? More general question: compute their minimal distance. A A - B B A and B Difference A - B distance (A,B)=distance(0,A-B) Principe of GJK algorithm : compute the distance between the origin O and B-A .
Algorithm GJK Principe of GJK algorithm : compute the distance between the origin O and B-A .
Algorithm GJK Principe of GJK algorithm : compute the distance between the origin O and B-A .
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